Manchester United boss Solskjaer wants to keep hold of Pogba, who has been heavily linked with a move to Real Madrid.
The World Cup winner impressed when Solskjaer succeeded Jose Mourinho in December but, as the season dragged on, his form tailed off.
Manchester United are adamant that Pogba will not be going anywhere, despite Real Madrid boss Zinedine Zidane talking up his compatriot in numerous press conferences.
And ESPN say that the France international will be offered the captaincy in order to ensure he stay.
Current skipper Antonio Valencia will leave the club as a free agent.
And Ander Herrera, who was previously tipped for the role during Mourinho’s time in charge, is also departing.
Ashley Young has spent much of the season as captain.
But ESPN say that, with United keen to sign a new right-back this summer, Solskjaer would rather give the armband to Pogba instead.
The Norwegian hopes that, by giving the midfielder more authority, he will return to his blockbuster best.
And Solskjaer is also hoping the armband will be enough to keep Real Madrid at bay.
Meanwhile, ESPN FC pundit Alejandro Moreno believes United should SELL Pogba – because it would benefit everybody involved.
“He has made it very clear, even though it is just little messages here and there, he has had ample opportunity to say ‘no, Manchester United is where I am at’, but instead he says ’there could be a possibility of another place that seems very interesting and it is a club of my dreams’,” Moreno said.
“Well, fine then, go to Real Madrid I say, I think they are in a position now where they know that they need talent and talented players that are either in the prime of their career or coming into their prime.
“There is no doubt that Paul Pogba is a world class talent but you need him to perform consistently.
“For whatever reason, and you can point the finger in all sorts of different directions to Mourinho and other people but it just hasn’t really worked out for him at Manchester United for whatever reason.
“And so maybe a change of scenery is a positive move for him and as for Real Madrid it does allow you to get your hands on a world class talent.”
Pogba has been at United since 2016.
The 26-year-old has won the EFL Cup and Europa League titles in the time since.